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reviewed

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Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Barry/barry601.html">On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.

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T(n,k) is the number of Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis) from (0,0) to (2n,0), having k peaks. Example: T(2,1)=3 because we have UU*DD, U*DH and HU*D, the peaks being shown by *. E.g., T(n,k) = binomial(n,k)*binomial(2n-k,n-1)/n for n>0. - Emeric Deutsch, Dec 06 2003

If C_n(x) is the g.f. of row n of the Narayana numbers (A001263), C_n(x) = Sum_{k=1..n}( binomial(n,k-1)*(binomial(n-1,k-1)/k ) * x^k) and T_n(x) is the g.f. of row n of T(n,k), then T_n(x) = C_n(x+1), or T(n,k) = [x^n]Sum_{k=1..n}(A001263(n,k)*(x+1)^k). - Mitch Harris, Jan 16 2007, Jan 31 2007

Consequently, with H(x,t) = 1/(dG(x,t)/dx) = (1+t+(2+t)*x+x^2)^2 / (1+t-x^2), the n-th A060693 polynomial in t is given by (1/n!)*[((H(x,t)*d/dx)^n] ) x evaluated at x=0, i.e., F(x,t) = exp[(x*H(u,t)*d/du] d) u, evaluated at u = 0.

00: [ 1]

01: [ 1, 1]

02: [ 2, 3, 1]

03: [ 5, 10, 6, 1]

04: [ 14, 35, 30, 10, 1]

05: [ 42, 126, 140, 70, 15, 1]

06: [ 132, 462, 630, 420, 140, 21, 1]

07: [ 429, 1716, 2772, 2310, 1050, 252, 28, 1]

08: [ 1430, 6435, 12012, 12012, 6930, 2310, 420, 36, 1]

09: [ 4862, 24310, 51480, 60060, 42042, 18018, 4620, 660, 45, 1]

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Krishna Menon and Anurag Singh, <a href="https://arxiv.org/abs/2212.13794">Grassmannian permutations avoiding identity</a>, arXiv:2212.13794 [math.CO], 2022.

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approved